diff --git a/packages/eslint-plugin/src/rules/no-floating-promises.ts b/packages/eslint-plugin/src/rules/no-floating-promises.ts
index fdc93cbed3f..466e29bbb9f 100644
--- a/packages/eslint-plugin/src/rules/no-floating-promises.ts
+++ b/packages/eslint-plugin/src/rules/no-floating-promises.ts
@@ -1,5 +1,6 @@
 import * as tsutils from 'tsutils';
 import * as ts from 'typescript';
+import { TSESLint } from '@typescript-eslint/experimental-utils';
 
 import * as util from '../util';
 
@@ -9,7 +10,9 @@ type Options = [
   },
 ];
 
-export default util.createRule<Options, 'floating'>({
+type MessageId = 'floating' | 'floatingVoid' | 'floatingFixVoid';
+
+export default util.createRule<Options, MessageId>({
   name: 'no-floating-promises',
   meta: {
     docs: {
@@ -20,6 +23,10 @@ export default util.createRule<Options, 'floating'>({
     },
     messages: {
       floating: 'Promises must be handled appropriately',
+      floatingVoid:
+        'Promises must be handled appropriately' +
+        ' or explicitly marked as ignored with the `void` operator',
+      floatingFixVoid: 'Add void operator to ignore',
     },
     schema: [
       {
@@ -41,16 +48,34 @@ export default util.createRule<Options, 'floating'>({
   create(context, [options]) {
     const parserServices = util.getParserServices(context);
     const checker = parserServices.program.getTypeChecker();
+    const sourceCode = context.getSourceCode();
 
     return {
       ExpressionStatement(node): void {
         const { expression } = parserServices.esTreeNodeToTSNodeMap.get(node);
 
         if (isUnhandledPromise(checker, expression)) {
-          context.report({
-            messageId: 'floating',
-            node,
-          });
+          if (options.ignoreVoid) {
+            context.report({
+              node,
+              messageId: 'floatingVoid',
+              suggest: [
+                {
+                  messageId: 'floatingFixVoid',
+                  fix(fixer): TSESLint.RuleFix {
+                    let code = sourceCode.getText(node);
+                    code = `void ${code}`;
+                    return fixer.replaceText(node, code);
+                  },
+                },
+              ],
+            });
+          } else {
+            context.report({
+              node,
+              messageId: 'floating',
+            });
+          }
         }
       },
     };
diff --git a/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts b/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts
index d381fb4ba7c..a6fcf777754 100644
--- a/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts
+++ b/packages/eslint-plugin/tests/rules/no-floating-promises.test.ts
@@ -2,7 +2,6 @@ import rule from '../../src/rules/no-floating-promises';
 import { RuleTester, getFixturesRootDir } from '../RuleTester';
 
 const rootDir = getFixturesRootDir();
-const messageId = 'floating';
 
 const ruleTester = new RuleTester({
   parserOptions: {
@@ -245,15 +244,39 @@ async function test() {
       errors: [
         {
           line: 3,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 4,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
+        },
+      ],
+    },
+    {
+      options: [{ ignoreVoid: true }],
+      code: `
+async function test() {
+  Promise.resolve("value");
+}
+`,
+      errors: [
+        {
+          line: 3,
+          messageId: 'floatingVoid',
+          suggestions: [
+            {
+              messageId: 'floatingFixVoid',
+              output: `
+async function test() {
+  void Promise.resolve("value");
+}
+`,
+            },
+          ],
         },
       ],
     },
@@ -268,15 +291,15 @@ async function test() {
       errors: [
         {
           line: 3,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 4,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -291,15 +314,15 @@ async function test() {
       errors: [
         {
           line: 3,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 4,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -316,15 +339,15 @@ async function test() {
       errors: [
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 6,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 7,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -338,11 +361,11 @@ async function test() {
       errors: [
         {
           line: 3,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 4,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -357,15 +380,15 @@ async function test() {
       errors: [
         {
           line: 3,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 4,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -378,7 +401,7 @@ async function test() {
       errors: [
         {
           line: 3,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -392,7 +415,7 @@ async function test() {
       errors: [
         {
           line: 4,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -405,7 +428,7 @@ async function test() {
       errors: [
         {
           line: 3,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -422,15 +445,15 @@ async function test() {
       errors: [
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 6,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 7,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -445,7 +468,7 @@ async function test() {
       errors: [
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -462,15 +485,15 @@ async function test() {
       errors: [
         {
           line: 5,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 6,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 7,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -488,15 +511,15 @@ async function test() {
       errors: [
         {
           line: 6,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 7,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 8,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -517,11 +540,11 @@ async function test() {
       errors: [
         {
           line: 10,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 11,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },
@@ -551,15 +574,15 @@ async function test() {
       errors: [
         {
           line: 18,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 19,
-          messageId,
+          messageId: 'floating',
         },
         {
           line: 20,
-          messageId,
+          messageId: 'floating',
         },
       ],
     },